Compact intracloud lightning discharges: 2. Estimation of electrical parameters

Authors


Abstract

[1] We estimated electrical parameters of 48 located compact intracloud discharges (CIDs) using their measured electric fields and vertical Hertzian dipole approximation. This approximation is consistent with the more general bouncing-wave CID model for a reasonably large subset of allowed combinations of propagation speed and channel length. For nine events, we estimated channel lengths from observed reflection signatures in measured dE/dt waveforms and assumed propagation speeds between 2 × 108 m/s and 3 × 108 m/s, which limit the range of allowed values. For a speed of 2.5 × 108 m/s (average value), the channel lengths for these nine events ranged from 108 to 142 m. The corresponding geometric mean values (GM) of peak current, zero-to-peak current risetime, and charge transfer in the first 5 μs are 143 kA, 5.4 μs, and 303 mC, respectively. The GM peak radiated power and energy radiated in the first 5 μs are 29 GW and 24 kJ, respectively. For the remaining 39 events, there were no reflection signatures observed, and the channel length was assumed to be 350 m, for which the Hertzian dipole approximation is valid for speeds in the range of 2 to 3 × 108 m/s. In this case, GM values of peak current, zero-to-peak current risetime, and charge transfer in the first 5 μs are 64 kA, 4.9 μs, and 142 mC, respectively. The GM peak radiated power and energy radiated in the first 5 μs are 28 GW and 32 kJ, respectively.

1. Introduction

[2] Nag and Rakov [2010] (part 1) showed that the current distribution along the CID channel is often not much different from uniform, because of relatively short channel length, Δh, relatively long current waveform, and relatively high propagation speed, v. This observation suggests that at least for some “allowed” combinations of v and Δh, we can reasonably approximate the CID channel by a Hertzian dipole. This approximation will enable us to simplify the field equations and use measured CID fields to infer various parameters of CIDs. Within the Hertzian dipole approximation, the propagation speed is not an input parameter and the current waveshape is the same as that of the time integral of CID radiation field signature.

[3] In this paper, we determine the limits of validity of the Hertzian dipole approximation as applied to CIDs, and use this approximation to infer the peak current, current risetime, charge transfer, radiated power, and radiated energy for the 48 located CIDs previously studied by Nag et al. [2010]. Additionally, we estimate the upper bound on cloud electric field prior to CID and total energy dissipated by this type of lightning discharge.

2. Hertzian Dipole Approximation

[4] The general time domain equation for computing the vertical electric field dEz due to a vertical differential current element idz (vertical dipole of length dz carrying a uniform current i(t)) at a height z above a perfectly conducting ground plane for the case of an observation point P being on the plane at a horizontal distance r from the dipole is given by [e.g., Uman, 1987; Thottappillil et al., 1997]

equation image

where ɛ0 is the electric permittivity of free space, tb(z) is the time at which the current is “seen” by an observer to begin at height z, c is the free-space speed of light, and R is the inclined distance from the dipole to the observation point, which is given by R(z) = equation image. From equation (1) for the geometry shown in Figure 1, the total electric field at the observation point for a finite-length channel whose lower and upper ends are at altitudes of h1 and h2, respectively, is given by

equation image

where h2 is a function of time during the first traversal of the channel and constant afterward.

Figure 1.

Geometrical parameters needed for calculating the vertical electric field at observation point P on perfectly conducting ground at horizontal distance r from the vertical CID channel.

[5] Equation (1) also applies to a vertical dipole of finite length Δh at height h, provided that Δh is very short compared to the shortest significant wavelength λ (Hertzian or electrically short dipole approximation). For example, a dipole of length, Δh = 500 m can be considered Hertzian if λ ≫ 500 m. This means that the above approximation is valid for frequencies f ≪ 600 kHz. For a vertical Hertzian dipole we can write

equation image

where R = equation image. Note that current i in equation (3) varies only as a function of time, with all the geometrical parameters being fixed.

[6] In section 3, we will show that the vertical Hertzian dipole approximation is consistent with the more general CID bouncing-wave model for a reasonably large subset of “allowed” combinations (established in part 1 [Nag and Rakov, 2010]) of v and Δh.

[7] Equation (3) can be rewritten as a second-order differential equation,

equation image

where arguments of Ez and i have been dropped to simplify notation. For known Ez and the geometrical parameters (Δh, h, and r) this equation can be numerically solved for i.

[8] We employed the Runge-Kutta method of order three (with four stages and an embedded second-order method, also known as the Bogacki–Shampine method [Bogacki and Shampine, 1989]) to solve equation (4) for i using measured electric fields Ez (which had a better signal-to-noise ratio than the measured electric field derivative waveforms) of 48 located CIDs that occurred at horizontal distances r ranging from 12 to 89 km and heights h ranging from 8.8 to 29 km [Nag et al., 2010]. The initial and final values of current were required to be zero, and the error tolerance of the numerical solution was set to 10−6. Channel lengths Δh for 9 CIDs were estimated from reflection signatures in electric field derivative (dE/dt) waveforms and assumed propagation speeds representing the entire range of their allowed values [see Nag and Rakov, 2010] (part 1). For the remaining 39 CIDs there were no reflection signatures observed, and a reasonable value of Δh = 350 m was assumed. This value is consistent with the Hertzian dipole approximations for speeds in the range of 2 to 3 × 108 m/s. We also considered other values of Δh, which are consistent with the Hertzian dipole approximation. Note that for Ez measured at far distances, the peak current can also be estimated using the radiation field approximation, given by

equation image

from which it follows that Ez is proportional to equation image. Equations (4) and (5) represent two ways to solve the “inverse source problem,” with equation (5) assuming that contributions from the electrostatic and induction field components (the first and second terms in equation (4)) are negligible. In order to compare equations (4) and (5), we (step a) assumed a CID current waveform, (step b) computed electric field at a horizontal distance of 200 km, where it is essentially radiation, using general equation (3), (step c) solved the inverse source problem using the computed electric field and equations (4) and (5), and (step d) compared the results with the assumed current. The computed electric field waveform at 200 km is shown in Figure 2a and the three current waveforms, the assumed one and those inferred from equations (4) and (5), are shown in Figure 2b. The bottom of the CID channel was assumed to be at 15 km and its length was set to 100 m. It is clear from Figure 2b that the three current waveforms are very similar to each other. For the 48 CIDs examined in this paper, occurring at horizontal distances ranging from 12 to 89 km, current peaks estimated using the radiation field approximation (equation (5)) were, on average, 15% greater than those obtained by solving exact equation (4). The differences in peak ranged from about 1% to 31% for 47 CIDs and for one CID the difference was 47%. We attributed the discrepancy to contributions from the electrostatic and induction field components at closer distances and to better constrained (regularized) solutions yielded by the Runge-Kutta procedure that required zero values for initial and final currents. We employed equation (4) for estimating CID currents presented in this paper.

Figure 2.

Comparison of equations (4) and (5). (a) Bouncing-wave-model-predicted electric field at a horizontal distance of 200 km for a CID with its channel bottom at a height of 15 km, Δh = 100 m, v = 2 × 108 m/s, ρ = 0, and the assumed input current shown in Figure 2b (black line). (b) Assumed current waveform (black line), used as an input in computing the electric field shown in Figure 2a, and those obtained by solving equations (4) (blue line) and (5) (red line). See text for details.

[9] In Appendix A, we estimated errors in current inferred from equation (5) due to uncertainties in CID heights and their horizontal distances from the field-measuring station [see Nag et al., 2010]. We assume that errors in current (due to uncertainties in the same two parameters) inferred from equation (4) are similar.

[10] The charge transferred up to time t can be obtained by integrating the current with respect to time,

equation image

Further, the source current can be used to find the radiation components of Eθ and Hϕ and hence the Poynting vector, total radiated power, and energy. The magnitude of the Poynting vector, which has the meaning of the radiated power density, can be obtained as

equation image

where Eθ = equation image sin θequation image and Hϕ = equation image sin θequation image [e.g., Uman, 1987]. After substituting the latter two field expressions in (7) we get

equation image

[11] The total radiated power, obtained by integrating (8) over a spherical surface of radius R, whose center is at the position of the dipole, is (presence of ground is not taken into account)

equation image

where θ and ϕ are the polar and azimuthal angles of the spherical coordinate system. Note that Prad is proportional to equation image2. The total energy dissipated up to time t can be obtained by integrating the radiated power with respect to time,

equation image

Since equation image is inversely proportional to Δh (see equation (5)), |equation image|, Prad, and W are each independent of Δh.

3. Limits of Validity of the Hertzian Dipole Approximation

[12] In this section, we compare electric fields produced at 200 km by a CID at a height of 15 km having a zero-to-peak current risetime RT of 6 μs computed using the vertical Hertzian dipole approximation with their counterparts predicted by the bouncing-wave CID model [see Nag and Rakov, 2010] (part 1). Calculations were performed for different combinations of effective current reflection coefficients at channel ends, channel lengths, and propagation speeds, each within the bounds (“allowed” ranges) established in part 1 [Nag and Rakov, 2010]. The Hertzian dipole was excited by the current found, using the bouncing wave model, for the middle of the channel (z = h1 + Δh/2) [see Nag and Rakov, 2010, Figure 4a] (part 1). The bouncing-wave model predicted waveforms were used as the ground truth, and electric field waveforms based on the Hertzian dipole approximation with initial peaks within about 15% of ground truth peaks were considered as confirming the validity of the approximation. The influence of RT was also considered.

[13] Figure 3 shows the electric fields for the Hertzian dipole approximation (dashed line) and bouncing-wave model (solid line) for ρ = 0, v = 2 × 108 m/s, RT = 6 μs, and channel lengths of 100, 350, and 700 m. While the Hertzian dipole approximation is acceptable for the lengths in the range of 100–350 m, it is not for 700 m. Figure 4 shows the electric fields for the Hertzian dipole approximation (dashed line) and bouncing-wave model (solid line) for ρ = −0.5, v = 2 × 108 m/s, RT = 6 μs, and channel lengths of 100 and 350 m. In both cases the Hertzian dipole approximation is acceptable. The results are summarized and compared with the “allowed” ranges of variation of v and Δh in Figure 5, from which one can see that the Hertzian dipole approximation is consistent with the bouncing-wave model for a reasonably large subset of “allowed” combinations of propagation speed and channel length. Specifically, it can be seen from Figure 5 that for ρ = 0 the Hertzian dipole approximation is valid for Δh ranging from about 100 to 550 m and for speeds ranging from about 0.7 × 108 m/s to 3 × 108 m/s, while the “allowed” ranges are from about 100 to 1000 m for Δh and from about 0.3 × 108 m/s to 3 × 108 m/s for v. For ρ = −0.5, the Hertzian dipole approximation is valid for the entire “allowed” domain (from about 100 to 500 m for Δh and from about 0.7 × 108 m/s to 3 × 108 m/s for v [see Nag and Rakov, 2010, Figure A4b] (part 1). Note that for RT = 6 μs the maximum value of Δh/v (channel traversal time), for which the Hertzian dipole approximation is acceptable, is almost constant and equal to about 1.9 μs and 1.7 μs for ρ = 0 and ρ = −0.5, respectively.

Figure 3.

Vertical electric fields at 200 km for the bouncing-wave model (solid line) for ρ = 0, v = 2 × 108 m/s, RT = 6 μs, and channel lengths of (a) 100, (b) 350, and (c) 700 m versus those for the Hertzian dipole approximation (dashed line). The Hertzian dipole approximation is acceptable for channel lengths in the range of 100–350 m, but not for 700 m.

Figure 4.

Same as Figure 3 but for ρ = −0.5 and channel lengths of (a) 100 and (b) 350 m, for both of which the Hertzian dipole approximation is acceptable.

Figure 5.

Comparison of the Hertzian dipole validity domain (combinations of propagation speed and channel length) with the “allowed” one for RT = 6 μs and ρ = 0. See text for details.

[14] Channel length values for which the Hertzian dipole approximation is valid (initial field peaks are within about 15% of bouncing-wave model predicted peaks) for different values of propagation speed and for RT = 6 μs are given in Table 1. Note that the errors for the opposite polarity overshoot were larger, so the Hertzian dipole approximation may be invalid after the initial half cycle, whose duration in our data set varies from 2.8 to 13 μs with a GM of 5.6 μs [Nag et al., 2010]. Since the errors at later times (during the opposite polarity overshoot of the electric field waveform) are larger, the waveforms for current, charge, radiated power, and energy (see, for example, Figure 6) may be invalid at those times. Additionally, for some of the electric field waveforms after the initial half cycle and particularly toward the tail of the opposite polarity overshoot the signal-to-noise ratio was rather poor. This is why we limited our estimates of charge transfer and energy to the initial 5 μs of the process. Note that the peak current, current risetimes, and peak power are generally unaffected by the uncertainties encountered at later times.

Figure 6.

(a) Measured electric field, (b) measured electric field derivative, (c) inferred current, (d) inferred charge transfer, (e) inferred radiated power, and (f) inferred radiated energy, each as a function of time, for a CID that occurred at a horizontal distance of 19 km, at an inferred height of 15 km, and had an inferred channel length of 138 m (v = 2.5 × 108 m/s). The current and power values given in the parentheses are the peaks, and the charge and energy values are estimated at 5 μs.

Table 1. Comparison of Electric Fields Based on the Hertzian Dipole Approximation and Bouncing-Wave Model for a Zero-to-Peak Current Risetime of 6 μs by Different Combinations of Currents Reflection Coefficients, Speeds, and Channel Lengthsa
Speed (m/s)Channel Length (m)Difference in Initial Peak (%)Difference in Opposite Polarity Overshoot (%)
  • a

    Combinations of speed and channel length in boldface are considered to be consistent with the Hertzian dipole approximation.

Reflection Coefficient ρ = −0.5
0.7 × 1081008.758
1251537
1 × 1081003.248
1701339
2 × 108100−0.76−1.9
3501539
3 × 108100−1.3−1.9
5001445
 
Reflection Coefficient ρ = 0
0.3 × 108100511.1
0.7 × 1081007.4−1.4
13515−0.77
250611.6
1 × 1081002.7−1.7
17011−0.7
19015−0.3
350601.6
2 × 108100−0.62−1.9
35013−1.0
37515−0.9
700631.8
3 × 108100−1.3−1.9
50012−1.1
55015−0.89
1000601.6

[15] We now consider the influence of zero-to-peak risetime, RT. Nag and Rakov [2010] (part 1) determined that the CID zero-to-peak current risetime is likely to be in the range from about 2 to 8.5 μs. For RT = 2 μs, the Hertzian dipole approximation is not valid for any allowed combination of parameters. For RT = 3 μs, the approximation is valid when v = 2 to 3 × 108 m/s and Δh = 100 to 200 m (channel traversal time, Δh/v = 0.3 to 1 μs) for both ρ = 0 and ρ = −0.5. For RT = 8.5 μs, the approximation is valid for Δh = 100 m and v = 1 to 3 × 108 m/s (channel traversal time, Δh/v = 0.3 to 1 μs), which cover the entire range of “allowed” values for both ρ = 0 and ρ = −0.5. In summary, the shorter the current risetime, the smaller the Hertzian dipole domain relative to the allowed one, as expected.

4. Electrical Parameters of CIDs

[16] Of the 48 located CIDs studied by Nag et al. [2010], for nine events, we were able to estimate channel lengths using reflection signatures in their measured dE/dt waveforms and assumed speeds within their limiting values of 2 × 108 and 3 × 108 m/s. Speeds lower than 2 × 108 m/s are not considered, since, within our model, for measured traversal times they would result in unreasonably small radiator lengths. For v = 2.5 × 108 m/s (average speed value), channel lengths for the nine events range from 108 to 142 m. These events occurred at horizontal distances of 19 to 89 km and estimated heights of 8.8 to 19 km, and had electric field peaks normalized to 100 km ranging from 14 to 35 V/m. Figure 6 shows the measured electric field and electric field derivative along with the inferred current, charge transferred, and radiated power and energy, each as a function of time, for one such event. Vertical arrows indicate reflection signatures in the dE/dt waveform. The estimated channel length (for v = 2.5 × 108 m/s) was 138 m. Table 2 summarizes the estimated parameters for all nine events. The geometric mean (GM) peak current is 143 kA with a range of 87 to 259 kA. The zero-to-peak current risetimes range from 3.0 to 9.5 μs with a GM of 5.4 μs. The charge transfer for the first 5 μs ranges from 79 to 496 mC with the GM being 303 mC. The GM peak radiated power and energy radiated for the first 5 μs are 29 GW (ranging from 12 to 70 GW) and 24 kJ (ranging from 7.5 to 52 kJ), respectively.

Table 2. Parameters of Nine Located CIDs With Channel Lengths Estimated Using Reflections in dE/dt Waveforms and Assumed Propagation Speed of 2.5 × 108 m/sa
Event IdentificationDistance (km)Radiator Height (km)Electric Field at 100 km (V/m)Radiator Length (m)Peak Current (kA)Zero-to-Peak Current Risetime (μs)10–90% Current Risetime (μs)Charge Transfer at 5 μs (mC)Peak Power (GW)Energy at 5 μs (kJ)
  • a

    AM, arithmetic mean; GM, geometric mean; Min, minimum value; and Max, maximum value.

082408_4271915141381407.93.92431817
083008_0098919351081973.81.54967052
083008_0315216281382595.93.44412839
083008_0455016301281423.41.54125040
083008_0525017301391483.01.44935950
091008_1383316141341189.54.979129.1
091008_1403613221421044.71.62722823
091108_1613010211231576.93.64123625
091108_176438.816113877.03.9230127.5
AM4514241291505.82.93423529
GM4114221291435.42.63032924
Min198.814108873.01.479127.5
Max8919351422599.54.94967052

[17] We now discuss uncertainties in the estimated electrical parameters for the nine events. For the lower bound on speed (2 × 108 m/s), all radiator lengths (86 to 114 m) are near the assumed lower bound, 100 m [Nag and Rakov, 2010, Appendix B] (part 1), which is expected because multiple reflection signatures are likely to be pronounced only for the shortest channels. For the upper bound on speed (3 × 108 m/s), the channel lengths would increase by a factor of 1.5 to range from 129 to 171 m. Since the inferred peak current and charge transfer are inversely proportional to channel length within the Hertzian dipole approximation (see section 2), these two parameters would decrease by a factor of 1.5. Thus, the uncertainty in our current and charge transfer estimates for the 9 events is ≤ 25% for the assumed v = 2.5 × 108 m/s. The current risetimes, peak power, and energy values, on the other hand, are independent of channel length (and hence of assumed speed) and will remain the same as those for 2.5 × 108 m/s. Table 3 summarizes the peak current and charge transfer at 5 μs each scaled to different channel lengths (inferred using measured channel traversal times and different assumed propagation speeds) for the nine CIDs.

Table 3. Peak Current and Charge Transfer at 5 μs Scaled to Different Channel Lengths That Were Inferred for Nine CIDs Using Reflection Signatures in Measured dE/dt Waveforms and Assumed Propagation Speeds
v (m/s)Δh (m)Peak Current (kA)Charge Transfer at 5 μs (mC)
AMGMMinMaxAMGMMinMaxAMGMMinMax
2 × 108 (lower bound)1031038611418817910832442737998621
3 × 108 (upper bound)1551541291711251197221628525365414
2.5 × 108 (average)1291291081421501438725934230379496

[18] Note that since all Δh values are less than 200 m (even for v = 3 × 108 m/s), all nine events can be reasonably approximated by Hertzian dipoles for almost the entire typical range of current zero-to-peak risetimes (about 2 to 8.5 μs). As a result, the inferred current risetimes (ranging from 3 to 9.5 μs with the GM being 5.4 μs) should be close to their true values, confirming our assumed typical zero-to-peak risetime of 6 μs.

[19] For the remaining 39 events, which did not exhibit reflection signatures, the electrical parameters were estimated for three assumed values of channel length, 170, 350, and 500 m, for which the Hertzian dipole approximation is valid (for RT = 6 μs) if the implied propagation speeds are ≥108, ≥2 × 108, and 3 × 108 m/s, respectively. CID electrical parameters for the three values of Δh are compared in Table 4. The geometric mean values of peak current for the assumed channel length values of Δh = 170 m, 350 m, and 500 m were 132, 64, and 45 kA, respectively. The corresponding geometric mean charge transfers at 5 μs were 293, 142, and 100 mC. Note that for the 39 events the Hertzian dipole validity domain is smaller than the “allowed” domain. As a result, we cannot assign any specific uncertainty to the estimated parameters in this case.

Table 4. Peak Current and Charge Transfer at 5 μs Scaled to Different Channel Lengths for 39 CIDs
Δh, mv (Allowed) m/sPeak Current (kA)Charge Transfer at 5 μs (mC)
AMGMMinMaxAMGMMinMax
170≥1081421326832832729345706
350≥2 × 10869643316015914222343
5003 × 10848452311211110015240

[20] Figures 7, 8, 9, 10, 11, and 12 show histograms for the peak current, zero-to-peak current risetime, 10−90% current risetime, charge transfer at 5 μs, peak radiated power, and energy at 5 μs, respectively, for all 48 events, including 9 events with channel lengths estimated from reflection signatures and 39 events with an assumed channel length of 350 m. The minimum, maximum, arithmetic, and geometric mean values for each parameter are given for the 9 and 39 events individually and for all 48 events combined. As noted earlier, current risetimes, peak radiated power, and radiated energy are independent of channel length, while peak current and charge transfer can be scaled to other channel lengths, provided that they are consistent with the Hertzian dipole approximation.

Figure 7.

Histogram of peak currents for 48 CIDs. For nine events (shaded histogram columns) with reflection signatures, channel lengths were inferred using channel traversal times measured in dE/dt waveforms and assumed propagation speed of 2.5 × 108 m/s. For the other 39 events (white histogram columns) an assumed channel length of 350 m (implied v ≥ 2 × 108 m/s) was used. Statistics given are the arithmetic mean (AM), geometric mean (GM), minimum value (Min), and maximum value (Max) for the 9 and 39 events individually and for all data combined.

Figure 8.

Histogram of zero-to-peak current risetimes for 48 CIDs. See also caption of Figure 7.

Figure 9.

Histogram of 10−90% current risetimes for 48 CIDs. See also caption of Figure 7.

Figure 10.

Histogram of charge transferred in 5 μs for 48 CIDs. See also caption of Figure 7.

Figure 11.

Histogram of peak radiated power for 48 CIDs. See also caption of Figure 7.

Figure 12.

Histogram of radiated energy in 5 μs for 48 CIDs. See also caption of Figure 7.

[21] For the 39 CIDs, the geometric mean values of zero-to-peak current risetime and 10−90% current risetime are 4.9 and 2.5 μs, respectively. The geometric mean peak radiated power and energy radiated for the first 5 μs are 28 GW and 32 kJ, respectively. For all 48 events, GM values of peak current, zero-to-peak current risetime, 10−90% current risetime, and charge transfer for the first 5 μs are 74 kA, 5.0 μs, 2.5 μs, and 164 mC, respectively. The corresponding geometric mean peak radiated power and energy radiated for the first 5 μs are 29 GW and 31 kJ. Note that in the distributions of peak current and charge transfer (Figures 7 and 10, respectively) the nine CIDs with reflection signatures constitute the tail of the histogram, which is related to their shorter inferred channel lengths. Overall, the CID current parameters are comparable to their counterparts for first return strokes in cloud-to-ground lightning, while their peak radiated electromagnetic power appears to be considerably higher, as discussed in section 7.

5. Upper Bound on Electric Field Prior to CID

[22] In this section, we will consider the CID as a discharge between two spherical charge regions. Such simplified configuration is similar to that employed by Cooray [1997] for studying regular cloud discharges. CIDs were observed to occur both inside clouds and above cloud tops [e.g., Smith et al., 2004; Nag et al., 2010]. The charge configuration considered in this section, as well as in section 6, is meant to apply to the in-cloud variety.

[23] For a CID current pulse with a peak current of 50 kA, total duration of 30 μs, and zero-to-peak risetime of 6 μs [see Nag and Rakov, 2010] (part 1) the total CID charge transfer (which is the time integral of current), Q, is 0.44 C. Let us consider two spherical volumes, one containing +0.44 C of charge and the other −0.44 C of charge, each with a uniform volume charge density, ρv, of 20 nC/m3 (absolute value) [e.g., Rakov and Uman, 2003, section 3.2.5]. Then, from Q = equation imageπb3ρv, the radius, b, of each sphere is 174 m. Let us assume that the vertical distance between the surfaces of these spherical regions is 100 m (assumed to be approximately equal to the assumed shortest CID channel length), as shown in Figure 13a, and that the medium outside the charged spheres contains zero net charge. Presence of ground and other charges in the cloud is neglected. The primary question to be answered here is: Can the positive and negative charges estimated for CIDs be accumulated 100 m apart without exceeding the conventional breakdown electric field? The 100 m assumption allows us to obtain the upper bound on electric field, since, all other conditions being the same, larger separations between the spheres will result in lower electric fields.

Figure 13.

(a) Simplified charge configuration giving rise to a CID. The CID channel length, Δh = 100 m, is assumed to be approximately equal to the distance between the two spherical charged regions. The electric field vector points upward and is assumed to be positive. (b) The electric field intensity as a function of z for x = 0 that is produced by each of the two charged regions (dashed line) shown in Figure 13a and the total electric field profile (solid line). Electric field, potential difference, and electrostatic energy values for this and other configurations are given in Table 5.

[24] For each sphere, the electric field intensity at any point at a distance r from the center of the sphere can be obtained using Gauss's law as

equation image

For the positively charged sphere ρv is positive and the electric field is directed radially outward, while for the negatively charged one it is negative and directed radially inward. For the configuration shown in Figure 13a, the total electric field at any point along the line joining the centers of the two spheres (x = 0) is given by superposition of the field contributions (both directed upward) from the two spheres. The total electric field intensity profile (solid line) and contributions from the individual spheres (dashed line) are shown in Figure 13b. Inside each of the charged spheres, the total electric field intensity increases almost linearly with distance from 2 × 104 V/m at the center to a maximum of 1.8 × 105 V/m at the surface. Outside the charged spheres, the electric field intensity is minimum (1.6 × 105 V/m) at a point equidistant from the two spheres. A CID channel is likely to be formed primarily between the two spherical volumes of charge, where the net electric charge is assumed to be zero and the electric field intensity is near its highest values. The charges are apparently collected in the source region and delivered to the vertical channel end via a heavily branched streamer network pervading this region. The wideband radiation field signature of such a “volume radiator” is presently unknown. We assume here that the observed wideband vertical electric field signatures of CIDs are essentially due to the current in the vertical channel. This assumption would be justified if the feeding streamers in the “source region” were predominantly horizontal (leaving aside the spherical geometry), because such channels would contribute little to nothing to CID vertical electric field waveforms.

[25] We will now examine dependence of electric field profile on charge transfer and volume charge density by using the peak currents for the nine events summarized in Table 2, but keeping the same current waveshape (zero-to-peak risetime of 6 μs and total duration of 30 μs). In section 4, we estimated the channel lengths for the nine located CIDs using channel traversal times measured in dE/dt waveforms and assumed propagation speed of 2.5 × 108 m/s to range from 108 to 142 m. The inferred peak currents ranged from 87 kA to 259 kA, with the geometric mean being 143 kA. The total charge transferred (over 30 μs current duration) for peak currents of 87, 143, and 259 kA are 0.77 C, 1.3 C, and 2.3 C, respectively. For a total charge transfer of 1.3 C and assumed volume charge densities of 2 nC/m3, 20 nC/m3, and 200 nC/m3, the radii of the spherical charge regions are 533 m, 247 m, and 115 m, respectively; the maximum electric field intensities (occurring between the spheres, on their surfaces) are 6.9 × 104, 2.8 × 105, and 1.1 × 106 V/m, respectively. The maximum electric field values for different combinations of charge transfer and assumed charge density are summarized in Table 5. Additionally given are minimum and average electric fields between the spherical charge regions and estimated electric potential difference between them. Except for the probably unrealistically high ρv = 200 nC/m3 cases (for which maximum fields range from 7.3 × 105 V/m to 1.4 × 106 V/m), the maximum electric field for all combinations of Q and ρv considered does not exceed 3.5 × 105 V/m, which is less than the conventional breakdown electric field in the cloud (on the order 106 V/m) and is generally on the order of 104 to 105 V/m. These maximum electric field estimates are for Δh = 100 m. All other conditions being the same, the maximum electric field will decrease with increasing Δh. The estimated electric fields appear to be too low for conventional breakdown, but may be sufficient for runaway electron breakdown involving an energetic cosmic ray particle.

Table 5. Maximum Electric Field at the Surface of Each of the Two Oppositely Charged Spheres Separated by a Zero Net Charge Region (See Figure 13a) for Different Combinations of Charge Transfer and Assumed Volume Charge Densitya
Peak Current (kA)Total Charge Transfer (Q) (C)Volume Charge Density (ρv) (nC/m3)Radius of Spherical Charge Regions (b) (m)Maximum Electric Field Between Spheres (Emax) (V/m)Minimum Electric Field Between Spheres (Emin) (V/m)Average Electric Field, Eav = (Emax + Emin)/2 (V/m)Potential Difference Between Spheres, ΔV ≈ EavΔh (V)External Electrostatic Energy, (QΔV) (J)Self (≈Internal) Electrostatic Energy (W1) (J)Mutual Electrostatic Energy (W2) (J)Total Energy (W = 2W1 + W2) (J)Total Energy (W = 2W1 + QΔV) (J)
  • a

    The distance Δh between the surfaces of charged spheres is assumed to be 100 m. Also given are the minimum and average electric fields, potential difference between the spheres, the total electrostatic energy and its components. See text for details.

500.4423754.6 × 1044.4 × 1044.5 × 1044.5 × 1062.0 × 1062.8 × 1062.1 × 1067.7 × 1067.6 × 106
  201741.8 × 1051.6 × 1051.7 × 1051.7 × 1070.7 × 1076.1 × 1063.9 × 1061.6 × 1071.9 × 107
  200817.3 × 1054.7 × 1056.0 × 1056.0 × 1072.6 × 1071.3 × 1076.7 × 1063.3 × 1075.2 × 107
87 (Min)0.7724515.7 × 1045.5 × 1045.6 × 1045.6 × 1064.3 × 1067.1 × 1065.3 × 1062.0 × 1071.9 × 107
  202102.3 × 1052.1 × 1052.2 × 1052.2 × 1071.7 × 1071.5 × 1071.0 × 1074.1 × 1074.7 × 107
  200979.1 × 1056.4 × 1057.8 × 1057.8 × 1076.0 × 1073.3 × 1071.8 × 1078.4 × 1071.3 × 108
143 (GM)1.325336.9 × 1046.7 × 1046.8 × 1046.8 × 1068.8 × 1061.6 × 1071.2 × 1074.5 × 1074.1 × 107
  202472.8 × 1052.6 × 1052.7 × 1052.7 × 1073.5 × 1073.5 × 1072.4 × 1079.4 × 1071.1 × 108
  2001151.1 × 1068.4 × 1051.0 × 1061.0 × 1081.3 × 1087.5 × 1074.4 × 1071.9 × 1082.8 × 108
259 (Max)2.326498.6 × 1048.4 × 1048.5 × 1048.5 × 1062.0 × 1074.4 × 1073.4 × 1071.2 × 1081.1 × 108
  203013.5 × 1053.3 × 1053.4 × 1053.4 × 1077.8 × 1079.4 × 1076.7 × 1072.6 × 1082.7 × 108
  2001401.4 × 1061.1 × 1061.3 × 1061.3 × 1083.0 × 1082.0 × 1081.2 × 1085.3 × 1087.0 × 108

[26] If the two spherical charge regions were in direct contact (Δh = 0) the total electric field profile would have maximum at the point of contact that is given by

equation image

where Δp = 2Qb is the dipole moment change for charge transfer Q over a distance of 2b. Smith et al. [1999] used such a configuration to impose a lower bound on CID channel length, which they assumed to extend between the centers of the charged spheres. In doing so, they computed Emax for their average measured dipole moment change, Δp = 0.38 C km, and different b and compared those field values with electric fields measured in thunderclouds. For b = 50 m (corresponds to their channel length of 100 m), they found Emax = 2.7 × 107 V/m, which is about an order of magnitude greater than the conventional breakdown electric field in the cloud, and concluded that such a short channel was physically unrealistic. Our estimated CID channel lengths on the order of 100 m, which are inferred from measured dE/dt waveforms, suggest that Smith et al.'s configuration, in which two oppositely charged regions are in direct contact, is inconsistent with at least some of our observations. In order to explain such short channel lengths, the charged regions should be separated by a region of essentially zero net charge (which can be created due to mixing) or by a region of very low charge density compared to that in the charged regions. The question of how exactly the charges are fed into the vertical CID channel remains open.

6. Total Energy Dissipated by CIDs

[27] In this section, we estimate the total energy dissipated by a CID with reference to the charge configuration shown in Figure 13a. If we assume that a CID neutralizes all the charge in each of the two spherical regions, the total energy dissipated will be equal to the total electrostatic energy stored in this charge configuration. One can find the total electrostatic energy as the sum of the electrostatic energies required to individually assemble the two uniformly charged spheres (self electrostatic energy [Cooray, 1997]) and the electrostatic energy due to the two spherical charge regions being placed in relatively close proximity to each other (mutual electrostatic energy).

[28] The self electrostatic energy, W1, of a uniformly charged sphere of radius b containing charge Q can be readily found as [Cheng, 1993]

equation image

This equation can be obtained by integrating the electrostatic energy density, equation image, where E is given by equation (11) for rb, over a spherical volume. Estimation of the mutual electrostatic energy, W2, is more complicated and requires additional simplifying assumptions. We will replace the two spherical charge regions with their equivalent point charges of magnitude +Q and −Q located at corresponding sphere centers, so that they are separated by distance (2b + Δh). This approximation is similar to that employed by Cooray [1997] for studying the energy dissipated by regular cloud discharges. The mutual electrostatic energy W2 of these two point charges is given by [Cheng, 1993]

equation image

Thus, the total electrostatic energy W of the overall charge configuration, and hence the energy dissipated by a resultant CID can be found as

equation image

[29] Estimates of energy dissipated by a CID having a 100 m long channel for different values of total charge transfer Q (computed for a fixed current waveshape and different peak currents) and volume charge density ρv are given in Table 5. The assumed volume charge density ρv is used to compute b in equation (13) for a given Q. For a charge transfer of 0.44 C (peak current of 50 kA), the total dissipated energy values are 7.7 × 106, 1.6 × 107, and 3.3 × 107 J for charge densities of 2, 20, and 200 nC/m3, respectively. For a charge density of 20 nC/m3 and charge transfers ranging from 0.44 C to 2.3 C the energy values range from 1.6 × 107 to 2.6 × 108 J. The corresponding average energy per unit length ranges from 1.6 × 105 to 2.6 × 106 J/m.

[30] It follows from equations (13) to (15) that, all other conditions being the same, the mutual electrostatic energy W2 due to the two equivalent point charges and hence the total electrostatic energy of the charge configuration shown in Figure 13a decreases with increasing the CID channel length. Thus, the values of total CID energy given above for the lower bound on channel length of 100 m should be viewed as the upper bound. For a CID current pulse with a peak of 50 kA, assumed ρv = 20 nC/m3, and channel lengths of 350, 500, and 1000 m, the total electrostatic energy values will be 1.47 × 107, 1.42 × 107, and 1.35 × 107 J, respectively, versus 1.6 × 107 J for Δh = 100 m.

[31] In our calculations of electrostatic energy, we have neglected the effect of ground. This simplifying assumption should not introduce a significant error because (1) CIDs occur at relatively large altitudes, typically greater than 10 km [Smith et al., 2004; Nag et al., 2010] and (2) W2 is inversely proportional to the distance between equivalent point charges. Indeed, the mutual electrostatic energy between a charge and its image separated by more than 20 km is at least an order of magnitude smaller than that between the two actual charges separated by less than 1.4 km (the limiting case in Table 5; Δh = 100 m). This is true even for CID channel lengths of 1000 m in which case the two equivalent point charges representing the actual charges are separated by less than 2.3 km.

[32] The total electrostatic energy of the charge configuration shown in Figure 13a can be alternatively estimated as the sum of energies stored inside the two spheres and energy stored between them. The internal energy can be computed as the volume integral of energy density, equation image, which is different from the self energy W1 in that E within each sphere should additionally include the contribution from the other sphere. As a zero approximation, we neglect this difference and assume that the internal energy is not much different from self energy W1 (the larger the separation between the spheres the smaller the difference). Further, we will roughly estimate the external energy as the product of charge transfer Q and potential difference ΔV between the surfaces of the two spheres, with ΔV being found as the average electric field intensity Eav between the spheres times the distance Δh between their surfaces. Values of Eav, ΔV, and the total energy, W = 2W1 + QΔV, are given in Table 5.

[33] The total electrostatic energy values estimated using the two different approaches are fairly similar, particularly for more realistic lower (2 and 20 nC/m3) values of ρv. As expected, the GM electromagnetic energy of 31 kJ radiated during the first 5 μs (see Figure 12) is much lower than the total CID energy estimates based on electrostatic considerations (see Table 5).

7. Discussion

[34] Smith et al. [1999], using distant (essentially radiation) electric field signatures of 15 CIDs from thunderstorms in New Mexico and west Texas, estimated the mean CID dipole moment change to be 0.38 C km. The minimum and maximum values were 0.26 and 0.80 C km. The mean dipole moment change duration (10–90%) for the 15 CIDs was 13.7 μs. They also inferred that CID channel lengths should be in the range of 300 to 1000 m. Using the mean dipole moment change (0.38 C km) and the limiting channel lengths we estimate the range of charge transfers for their CIDs to be 0.38 C to 1.27 C. Our estimated charge transfers for the first 5 μs are on the order of tens to hundreds of millicoulombs (see Tables 2, 3 and 4 and Figure 10) and are apparently consistent with Smith et al.'s values which correspond to the mean charge transfer time of 13.7 μs.

[35] Eack [2004], using simultaneous measurements of electric fields at near and far distances for seven CIDs, estimated that, on average, a CID transferred 0.3 C over a distance of 3.2 km. These results are based on a value of speed derived from the misinterpreted dipole approximation equation [see Nag and Rakov, 2010, section 8] (part 1) and, therefore, are invalid. For one event produced by a New Mexico thunderstorm, Eack also estimated the peak current of 29 kA from the measured predominantly induction electric field peak (assuming a uniform current over the entire channel length). For the same event, Watson and Marshall [2007], using the transmission line model, inferred a considerably larger peak current of 74 kA, which is about the same as our estimates based on both the bouncing-wave model and Hertzian dipole approximation [see Nag and Rakov, 2010] (part 1).

[36] For downward negative lightning, Berger et al. [1975] reported a median total charge transfers of 5.2 C and 1.4 C for first and subsequent strokes, respectively. Schoene et al. [2009] reported the geometric mean charge transfer within 1 ms after the beginning of the return stroke for 151 negative rocket-triggered lightning strokes (which are similar to subsequent strokes in natural lightning) to be 1.0 C. Using in situ measured electric field profiles, Maggio et al. [2009] estimated the average (probably arithmetic mean) charge transferred by 29 IC flashes to be 17.6 C. Thus, charges transferred by individual CG strokes and by regular ICs are considerably larger than those transferred within the first 5 μs by CIDs.

[37] Rakov et al. [1998], from two-station measurements of electric and magnetic fields of a dart-stepped leader in triggered lightning, estimated that the formation of each leader step is associated with a charge of a few millicoulombs and a current of a few kiloamperes. From measurements of electric field pulses radiated by leader steps of first strokes in natural lightning, Krider et al. [1977] inferred that the minimum charge involved in the formation of a step is 1–4 mC and the peak step current is at least 2–8 kA. The charge transfers for cloud-to-ground lightning leader steps are 1−2 orders of magnitude smaller than those transferred during the first 5 μs by CIDs. Our estimated CID peak currents of tens to hundreds of kiloamperes are at least an order of magnitude greater than those expected for leader steps. Note that the step-formation process is thought to occur on a time scale on the order of 1 μs, and typical step lengths are 10 and 50 m for dart-stepped and stepped negative leaders, respectively [Rakov and Uman, 2003].

[38] Krider et al. [1968] estimated the peak input power per unit channel length for a natural lightning first stroke to be 7.8 × 108 W/m. Jayakumar et al. [2006] found the mean value of peak input power per unit length for triggered lightning strokes to be 9.6 × 108 W/m. Krider and Guo [1983] and Krider [1992] estimated the wideband radio-frequency electromagnetic power radiated by a subsequent return stroke at the time of the electric field peak to be 3 to 5 GW. The average zero-to-peak risetime of the subsequent stroke field waveforms was 2.8 μs, so that the radiating channel length at the time of field peak was probably some hundreds of meters, which is comparable to the channel length for CIDs. Our arithmetic mean peak radiated power (wideband) of 37 GW found for 48 CIDs is about an order of magnitude higher than the above estimates for subsequent return strokes. For first strokes, Krider and Guo [1983] estimated an arithmetic mean peak electromagnetic power of 20 GW, which is still lower than our estimate for CIDs.

[39] Rison et al. [1999] and Thomas et al. [2001] reported the peak radiated power in the narrowband VHF (60–66 MHz) frequency range of the Lightning Mapping Array (LMA) for one New Mexico CID to be greater than 100 kW and greater than 300 kW, respectively. Source peak powers radiated at 60–66 MHz by “normal” lightning processes ranged from 1 W (minimum locatable value) up to 10–30 kW [Thomas et al., 2001]. Rison et al. [1999] stated that the peak VHF radiation from CIDs was typically 30 dB (a factor of 1000) stronger than that from other lightning processes.

[40] Our estimates of CID energy per unit channel length (based on values in the next to last column of Table 5 and Δh = 100 m) range from 7.7 × 104 to 5.3 × 106 J/m. Interestingly, the input energy per unit length for a natural lightning first stroke of 2.3 × 105 J/m reported by Krider et al. [1968] is within this range. For triggered lightning strokes, Jayakumar et al. [2006] estimated the mean value of input energy per unit length for 36 triggered lightning strokes to be 3.6 × 103 J/m, which is more than an order of magnitude smaller than the lower bound estimated here for CIDs.

[41] Cooray [1997] estimated the energy dissipated by a typical first leader/return stroke sequence to be 5.5 × 108 J for a 5 km long channel, which corresponds to 1.1 × 105 J/m. Assuming that the first stroke channel length ranges from 5 to 8 km [e.g., Rakov and Uman, 2003] and Cooray's value of energy per unit length, we estimated the total energy range to be from 5.5 to 8.8 × 108 J. Cooray [1997] estimated the dissipated energy for regular cloud discharges to be about 18 × 108 J for a charge transfer of 8 C. The corresponding channel length was 2.5 km. Taking the charge transfer of 8 C as typical for cloud discharges we estimate the energy per unit channel length to be 7.2 × 105 J/m. Using this latter value and the range of IC channel lengths of 2 to 5 km [e.g., Shao and Krehbiel, 1996], we estimate the IC energy range of 1.4 to 3.6 × 109 J. Maggio et al. [2009], using median charge transfers and average in-cloud potentials, estimated the energy dissipated by 16 IC flashes to be 1.5 × 109 J per flash, which is near the lower bound of our range for regular ICs. Channel lengths for CIDs are within the range of 100 m to 1000 m [see Nag and Rakov, 2010] (part 1). As seen in Table 5, the energies dissipated by a CID with Δh = 100 m and ρv = 20 C/m, transferring 1.3 C of charge, is about 108 J, or 106 J/m. Using this energy per unit length and the range of channel lengths, we estimate the range of values of total energy dissipated by CIDs to be 108 to 109 J. Note that the upper end of this range is similar to the energy dissipated by a typical multiple-stroke ground flash [Rakov and Uman, 2003].

[42] Figure 14 summarizes the ranges of total energy dissipated by first strokes in cloud-to-ground discharges, regular intracloud discharges, and CIDs along with corresponding ranges for channel length. It appears that CIDs typically do not surpass either cloud-to-ground first strokes or regular cloud discharges in terms of energy. Hence, the labels like “energetic intracloud discharges” [e.g., Eack, 2004; Smith et al., 2004] and “high-energy discharges” [Watson and Marshall, 2007] that are sometimes used to refer to CIDs are probably not justified. As seen in Figure 14, the most distinctive feature of CIDs (besides their very intense HF-VHF radiation) is their small spatial extent.

Figure 14.

Ranges of dissipated energy and channel length for typical first strokes in cloud-to-ground flashes (CGs), regular intracloud flashes (ICs), and CIDs; charge transfers were assumed to be 5 C, 8 C, and 1.3 C, respectively.

8. Summary

[43] Compact Intracloud Discharges (CIDs) are either in- or above-cloud lightning discharges that produce single bipolar electric field pulses having typical full widths of 10 to 30 μs and intense HF-VHF radiation bursts (much more intense than those from any other cloud-to-ground or “normal” cloud discharge process). We estimated electrical parameters of 48 located CIDs using their measured electric fields and the vertical Hertzian dipole approximation. This approximation is consistent with the more general bouncing-wave model for a reasonably large subset of allowed combinations of propagation speed and channel length. For example, for zero-to-peak current risetime RT = 6 μs and propagation speed v = 2 × 108 m/s the allowed range of channel length, Δh, is from 100 to 700 m, with the Hertzian dipole approximation being valid for Δh up to 375 m.

[44] For nine events, we estimated CID channel lengths from channel traversal times measured in dE/dt waveforms and assumed propagation speeds of 2 × 108 m/s to 3 × 108 m/s, which limit the range of allowed speed values. For v = 2.5 × 108 m/s (average value), the channel lengths for these nine events ranged from 108 to 142 m. The corresponding geometric mean values of peak current, zero-to-peak current risetime, 10−90% current risetime, and charge transfer for the first 5 μs are 143 kA, 5.4 μs, 2.6 μs, and 303 mC, respectively. The uncertainty in our current and charge transfer values for the nine events was estimated to be ≤ 25%. The geometric mean peak radiated power and energy radiated for the first 5 μs (both wideband) are 29 GW and 24 kJ, respectively.

[45] For the remaining 39 events, there were no reflection signatures observed, and Δh was assumed to be 350 m, for which the Hertzian dipole approximation is valid for speeds in the range of 2 to 3 × 108 m/s. In this case, geometric mean values of peak current, zero-to-peak current risetime, 10−90% current risetime, and charge transfer for the first 5 μs are 64 kA, 2.5 μs, and 142 mC, respectively. The geometric mean peak radiated power and energy radiated for the first 5 μs are 28 GW and 32 kJ, respectively.

[46] The radiated power and energy are independent of Δh, while peak current and charge transfer can be scaled to other channel lengths, provided that they are consistent with the Hertzian dipole approximation. For the 39 events, we additionally considered Δh = 170 m, which is consistent with v ≥ 108 m/s, and Δh = 500 m, which is consistent with v = 3 × 108 m/s.

[47] For all 48 events, GM values of peak current, zero-to-peak current risetime, 10−90% current risetime, and charge transfer for the first 5 μs are 74 kA, 5 μs, 2.5 μs, and 164 mC, respectively. The geometric mean peak radiated power, and energy radiated for the first 5 μs are 29 GW and 31 kJ, respectively. Overall, the estimated CID current waveform parameters are comparable to their counterparts for first strokes in cloud-to-ground lightning, while their peak radiated electromagnetic power appears to be considerably higher.

[48] The upper bound on electric field prior to CIDs is generally on the order of 104 to 105 V/m. The total energy dissipated by a CID typically ranges from 108 to 109 J, which is comparable to the energy dissipated by a typical multiple-stroke ground flash and lower than that dissipated by a regular cloud discharge.

Appendix A:: Errors in CID Electrical Parameters Due to Uncertainties in CID Locations

[49] Uncertainties in CID locations include errors in heights, h, and horizontal distances, r, from the field measuring station. Nag et al. [2010] estimated errors in heights of 48 CIDs considered here to range from 4.7 to 95% with a mean of 17%. Most of the events (42 of 48) had height errors less than 30% and 39 had errors less than 25%. They found that the error in source height is a sensitive function of equation image when this ratio exceeds 0.95 (α < 18°) or so [see Nag et al., 2010, Figure A1]. Horizontal distances to the 48 CIDs from the field-measuring station were estimated using NLDN data. Errors in r ranged from 0.63 to 21%. In this Appendix, we examine the effect on the inferred CID parameters of errors in h and r. We will show that, with one exception, errors in CID currents due to errors in h and r do not exceed 15%, even for events whose height errors are in the 30 to 95% range.

[50] One can see from equation (5) that Ez is proportional to equation image, the proportionality constant being equation image, where equation image. Any error in estimation of i from Ez due to uncertainties in CID location is associated with the ratio equation image, which is a function of two variables, h and r. The error in equation image was estimated from errors in r and h, using the derivative method [Taylor, 1997], which was also employed by Nag et al. [2010] to estimate errors in CID heights. The error in equation image was defined as equation image = |equation image| equation image + |equation image|equation image, and values of equation image and equation image for the 48 CIDs were taken from Nag et al. [2010].

[51] The computed errors in equation image versus errors in h and r are shown in Figures A1 and A2, respectively. One can see in Figure A1 that even though errors in height for six events are in the 30 to 95% range, errors in equation image are less than 15% with one exception for which the error is 23%. These results can be explained as follows. Events with larger (up to 95%) height errors correspond to larger horizontal distances r, so that R only weakly depends on h. Further, errors in r are relatively small, mostly less than 5% (see Figure A2). As a result, relatively large errors in h do not lead to large errors in the ratio equation image which is needed to find i from Ez. The outlier in each of Figures A1 and A2, which is characterized by the largest error in equation image of 23%, is associated with an unusually large error in r.

Figure A1.

Errors in ratio equation image which is needed in solving the inverse source problem (finding i from measured Ez) versus errors in CID heights.

Figure A2.

Same as Figure A1 but versus errors in the horizontal distances to CIDs from the field-measuring station.

[52] The error analysis above applies to equation (5), but we assume that errors in currents inferred from equation (4), due to uncertainties in CID locations, are similar. We conclude that the errors in h and r are not expected to have a significant effect on the inferred parameters of the 48 CIDs considered in this paper.

Acknowledgments

[53] The authors would like to thank D. Tsalikis for his help in developing instrumentation and performing measurements and J. Cramer of Vaisala for providing NLDN data. This research was supported in part by NSF grants ATM-0346164 and ATM-0852869 and by DARPA.

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